Marginal profit

Question

If the total profit, in thousands of rands, for a product is given by the function: P(x) = 20√

x + 1 − 2x.

What is the marginal profit function?

What is the marginal profit at a production level of 15 units?

Answer 1

Yolanda,

You asked –

If the total profit, in thousands of rands, for a product is given by the function: P(x) = 20√

x + 1 − 2x.

What is the marginal profit function?

What is the marginal profit at a production level of 15 units?

Answer:

I’ll assume when you state marginal profit – you mean the break-even point, where your profit is zero.  To find the number of units to break-even we set your profit function to 0 and solve for x, the number of units.

0 = 20*sqrt(x) + 1 – 2x; to solve we are going to transform this equation, let y = sqrt(x)

20y+1-2y^2 = 0; rearrange to put in standard form

-2y^2+20y+1 = 0; solve this trinomial using the quadratic equation

Y = -0.050, 10.05 (rounded); note to solve for y we use our transformation

X = y^2; to find x from our transform we square both sides, then substitute y

X = 0.0025, 100.0025; now because we used a transform for a sqrt function, we need validate these answers in the original equation

Using x=0.0025, we find that P(x) = 2 so that is not a valid answer

Using x = 100.0025, we find that P(x) = 0 so that is a VALID answer

Looking at a graph of this function you’ll find that the profit maximizes at 25 units and then goes negative after 100 units.

At 15 units, the profit is approximately 48 rands. We get this by substituting into the original equation.

Regards,

Ray